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Physics Factsheet April 2003

Number 51

Electromagnetic Doppler effect and the expanding Universe This Factsheet will explain: •. The theory of the Doppler effect as it affects electromagnetic waves; • Emission and absorption spectra; • The red shift of radiation from distant galaxies and Hubble’s Law; • How evidence of the Doppler effect in absorption spectra gives us knowledge of the Universe and its origins.

Fig 1. Frequency of e-m radiation from a receding object

1 2 3

Before studying this Factsheet, you should be familiar with: • ideas of the “Red Shift” and the “Big Bang” theory from the GCSE syllabus.

4 blueshift

Emission Spectra The spectra of hot gases can be observed using a diffraction grating.

You should also be familiar with the idea of the energy levels in an atom, and with the quantization relationship (Factsheet 1):

Hot gases emit a spectrum, but it is not a continuous spectrum like a rainbow (called a “black body spectrum”). It consists of discrete lines as shown in the diagram.

where: ∆E = energy level gap h = Planck’s constant f = frequency of the emitted radiation

Li

The Doppler Effect

Na

You will have noticed the Doppler effect in sound waves: when a train approaches a station, the pitch (and therefore the frequency) of its whistle rises until it passes an observer on the platform, then falls as the train goes past the observer and moves away. If you hurl a whistle around your head on the end of a piece of string you can hear the changing frequency as the whistle approaches and recedes from your ear.

K Rb Cs Hg

The same effect occurs in electromagnetic waves. The explanation lies in the fact that, unlike a ball thrown from a moving train, the speed of e-m radiation is constant no matter what the speed of the object from which it originates.

Ne 400

Frequency of e-m radiation from a receding object Consider fig 1. Waves from the object travel with a speed c (3.0 × 10 ms ), the object is receding with a speed of v, so the waves become spread out over a greater distance, i.e. their wavelength increases. If their wavelength increases while their speed stays the same, then the frequency decreases. The change in wavelength ∆λ, causes a change in frequency ∆f. 8

It can be shown that

F2 F1

If radiation is emitted from a receding source, its frequency is ∆λ ∆f v = = lowered slightly. The relationship is given by: f c λ

You should be familiar with principle of superposition of light, the ideas of diffraction and the use of a diffraction grating to produce a spectrum.

∆E = hf

redshift F4 F3

You should also know: • The relationship: v = fλ, which connects the velocity, frequency and wavelength of a wave. Factsheet No.17 explores the properties of waves and this relationship. • That the electromagnetic spectrum is a family of waves, which all travel at the same velocity – the so-called “velocity of light” (3.0 × 108 ms-1), but with different wavelengths; this ranges from waves with the longest (radio waves) with a wavelength of the order of km, to those with the shortest (γ-rays) with a wavelength of the order of pm (10-12m). • That light (and therefore all the other radiation in the e-m spectrum) is observed to travel at 3.0 × 108 ms-1 no matter what the speed of the object which is emitting the radiation.

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500 600 wavelength in nanometers

700

When the gas is heated, the electrons are excited into a higher energy state. When they drop back into lower energy states they emit a photon of radiation of a specific frequency. The energy of this photon is given by:

∆λ ∆f v = = f c λ

f=

∆E where f = the frequency h ∆E = the energy level gap h = Planck’s constant

This gives a line of a frequency characteristic of the element which is emitting the radiation.

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Physics Factsheet

Electromagnetic Doppler Effect and the Expanding Universe Unit of distance

An emission spectrum consists of discrete lines of frequency characteristic of the elements emitting the radiation

Since distances in space are so vast they would become unmanageable in m or km. Several other units are used. One is the light-year. A light-year is the distance travelled by light in a year.

Example 1 Calculate the frequency of the emission line in a hydrogen spectrum, which corresponds to a transition between energy levels of ∆E = 10.19eV. Planck’s constant h = 6.6 × 10-34Js; e = 1.6 × 10-19C E 1.6 × 10-19 × 10.19 f= = = 2.47 × 1015Hz h 6.6 × 10-34

1 light-year = 3 × 108 × 60 × 60 × 24 × 365.25 = 9.467× 1015m

The Big Bang A sensible explanation for the red shift of light from distant galaxies is that they are receding. This supports the Big Bang theory of the origin of the Universe. If the entire Universe began as a concentrated point and then exploded, then we would expect all galaxies to be moving away from each other. Additionally we would expect them to be slowing down, since the only force acting should be the gravitational pull of other galaxies, to reduce the initial velocity after the bang.

Absorption Spectra When “white light”, i.e. radiation of all frequencies, passes through cooler gases, radiation is absorbed at the frequencies where it would have been emitted by the hot gases. So radiation passing through the cooler layers of the atmosphere of distant stars acquires dark lines characteristic of the gases in the atmosphere. Thus we can recognize, say, hydrogen and helium lines in these absorption spectra from distant stars.

Hubble’s Law suggests that those galaxies further away from Earth are moving faster. The furthest galaxies are also the oldest, since the light takes longer to reach us, so we are seeing them as they were long ago. Thus the older galaxies are travelling faster.

An absorption spectrum consists of a “white light” background crossed with dark lines of frequencies characteristic of those elements which have absorbed the radiation.

The red shift of galactic radiation supports the Big Bang Theory Doppler shift of lines in absorption spectra. Studying the absorption spectra of the light from distant stars does indeed show characteristic lines, but the frequencies are shifted by a fixed amount to slightly lower values than those observed close to Earth. (The so-called “Red shift”, since the red end of the visible spectrum is of lower frequency than the violet end.) So, for example, if there was a series of hydrogen lines, they would have the same difference in frequencies as those observed with hydrogen spectra from sources on Earth but they would be of slightly lower frequency.

The Age of the Universe Hubble’s Law can also be used to give an approximate value for the age of the Universe. If a galaxy is distance d from our own, and has a recession velocity of v, then separation must have occurred at a time d ago. v This represents the approximate age of the Universe. d 1 = v H This gives an approximate age of the Universe as 1 – 2 × 1010 years (10 – 20 billion years). There are two problems with this: (a) it assumes a constant speed, and as we have suggested, the speed is thought to have decreased from an initial value to its present value; (b ) the large uncertainty in H. These factors mean that this is a very approximate estimate! Since v = Hd ,

As explained above, the Doppler effect could account for this red shift, if the source of the radiation is moving away from Earth. It is possible to calculate the speed at which the source would need to be moving away in order to give a particular frequency shift, from the formula given above, v ∆λ = λ c

Typical Exam Question (a) The table lists three physical quantities. Complete the table.

The Doppler effect can explain the shift to lower frequencies, which is observed in the characteristic lines in absorption spectra of the radiation from distant stars. The stars are moving away from each other or receding.

Physical Quantity The Hubble constant Galactic distances

Example 2 Calculate the speed at which a source would be receding, if a line in its spectrum was shifted from a wavelength of 600nm, to 603nm. ∆λ v 3 × 10-9 = so v = × 3 × 108 = 1500 km s−1 λ c 600 × 10-9

Planck’s constant [6] (b) Write a short paragraph explaining how the Hubble constant gives a value for the approximate age of the Universe. [4]

Hubble’s Law The astronomer Edwin Hubble calculated the speeds at which observed galaxies were receding, from the red shift of their absorption spectra . He looked for a relationship between the speeds and the distance away of the galaxy and found that the speed is directly proportional to the distance away. This is expressed as Hubble’s Law:

Answer (a) Physical Quantity

v = Hd Where: d = the distance of the galaxy from Earth v = speed in compatible units H = the Hubble constant (thought to lie in the range 1.6 - 2.3 × 10-18 s-1) Large distances are very difficult to estimate accurately, so there is a large uncertainty in the value of H. (These methods are beyond the scope of A2 study, though methods of estimating distances to the nearest galaxies are dealt with in the Astrophysics Option of some specifications.)

Any commonly used unit Base units

Any commonly used unit

Base units

The Hubble constant

kms-1Mpc-1 or s-1

s-1

Galactic distances

light-year

m

Planck’s constant

Js

kgm2s-1 *

*(energy = force × distance, force is mass × accel, accel is ms-2)

(b) Hubble’s Law relates the distance away of distant galaxies with their speed of recession as calculated from the Doppler shift of their radiation, by the equation: v= Hd

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Physics Factsheet

Electromagnetic Doppler Effect and the Expanding Universe Exam Workshop

Questions

This is a typical poor student’s answer to an exam question. The comments explain what is wrong with the answers and how they can be improved. The examiner’s answer is given below.

1. Explain why radiation from a moving source experiences a shift of wavelength. 2. Give the equation for the shift of wavelength.

(a) The Doppler shift may be used to study the movement of distant galaxies. Explain what is meant by a Doppler shift and how it is used to study the movement of distant galaxies. You may be awarded a mark for the clarity of your answer. [5]

3. Is light from a receding source shifted to longer or shorter wavelengths? 4. State Hubble’s Law. 5. A spectral line in the absorption spectrum of a distant galaxy shows a Doppler shift from 550nm to 555nm. (a) Calculate the speed of recession of the galaxy. (b) If the galaxy’s distance away is estimated to be 1.44 × 1021km, what value does this give for the Hubble constant?

Doppler shift is when the wavelength is shifted. It shows that galaxies are moving apart. This supports the Big Bang theory. 1/5 The student has been award a mark for mentioning movement away associated with a shift of wavelength, but has failed to identify what wavelength s/he is talking about, or which way it is shifted. No explanation of the Doppler effect has been given. No credit has been given for the reference to the Big Bang theory, since this is not called for in the question.

Answers 1. See text 2. See text 3. Longer 4. v = Hd 5. a) ∆λ v 5 × 3 × 108 = so v = = 2.7 ×106 m/s λ c 550

(b) Hubble’s law states that: v = Hd where d is the distance of a galaxy from Earth, and v its speed in compatible units, and H is the Hubble constant. (i) Give a suitable unit for H [1] v is in light-years and d in m, so H is light-years/m

b)

0/1

v = Hd, so H =

v 2.7 × 106 = = 1.9 × 10-18 s-1 d 1.44 × 1024

The student has become confused. The light-year is a unit of distance, not time or speed. (ii) Explain why there is a large uncertainty in H because the quantities are difficult to measure accurately

[2] 1/2

While the student appreciates that difficulties in measuring cause an uncertainty, s/he has not been sufficiently specific that it is the DISTANCE which is difficult to measure with accuracy. (iii) Explain how the Doppler shift and Hubble’s Law support the Big Bang theory. [2] Galaxies are moving away from each other, so must have exploded in the past. 1/2 Not sufficient for both marks.

Examiner’s Answers (a) When EM radiation is emitted from a moving ! source, the discrete spectral lines in its absorption spectrum suffer a change of wavelength.

!

! If the source is moving away, then the shift is to longer wavelengths. The speed of recession can be calculated from ∆λ = v ! ! λ c −1 (b) (i) If d is in km, then compatible units for v would be km s , so units of H are s-1. ! !

(ii) To determine H requires measurement of λ,∆λ, and d. Measurement of λ and ∆λ can be done with reasonable accuracy, but d is very difficult to measure accurately.

!

(iii) The galaxies that are furthest away are also the oldest, since light has taken longest to reach us. ! These are moving faster, so younger, closer galaxies are moving at slower speeds, as would be expected if they had been produced in a huge explosion at some stage in the past.

!

Acknowledgements: This Physics Factsheet was researched and written by Janice Jones. The Curriculum Press,Unit 305B, The Big Peg,120 Vyse Street, Birmingham, B18 6NF. Physics Factsheets may be copied free of charge by teaching staff or students, provided that their school is a registered subscriber. No part of these Factsheets may be reproduced, stored in a retrieval system, or transmitted, in any other form or by any other means, without the prior permission of the publisher. ISSN 1351-5136

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